cp's OEIS Frontend

A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

%I A195692 #37 Sep 03 2025 07:42:37
%S A195692 9,0,4,5,5,6,8,9,4,3,0,2,3,8,1,3,6,4,1,2,7,3,1,6,7,9,5,6,6,1,9,5,8,7,
%T A195692 2,1,4,3,1,0,9,4,5,6,0,9,6,1,6,0,5,0,6,7,6,5,5,9,9,8,4,5,3,3,4,9,9,2,
%U A195692 9,2,1,3,7,6,4,0,4,5,2,1,7,6,0,6,1,1,0,5,8,1,5,0,1,4,7,7,3,9,8,7,3,1,2,9,7
%N A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).
%C A195692 Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - _Michel Marcus_, Aug 13 2019
%C A195692 The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - _Amiram Eldar_, Jan 22 2022
%C A195692 Smallest positive root of the equation sin(x) = cot(x). - _Wolfe Padawer_, Apr 11 2023
%H A195692 Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, <a href="https://arxiv.org/abs/1908.02251">Grid dissections of tangential quadrilaterals</a>, arXiv:1908.02251 [math.MG], 2019.
%H A195692 Duane W. DeTemple, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/30-3/detemple.pdf">The Triangle of Smallest Perimeter which Circumscribes a Semicircle</a>, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
%H A195692 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195692 From _Amiram Eldar_, Feb 07 2022: (Start)
%F A195692 Equals Pi/2 - A175288.
%F A195692 Equals arcsin(1/sqrt(phi)).
%F A195692 Equals arctan(sqrt(phi)). (End)
%e A195692 arccos(1/phi) = 0.904556894302381364127316795661958721...
%e A195692 cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
%e A195692 sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...
%t A195692 r = 1/GoldenRatio;
%t A195692 N[ArcCos[r], 100]
%t A195692 RealDigits[%]
%o A195692 (PARI) acos(2/(sqrt(5)+1)) \\ _Charles R Greathouse IV_, Nov 21 2024
%Y A195692 Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.
%K A195692 nonn,cons,changed
%O A195692 0,1
%A A195692 _Clark Kimberling_, Sep 22 2011
%E A195692 Terms replaced with intended terms by _Rick L. Shepherd_, Jan 30 2013